If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and a – b is divisible by 5}. Prove that R is an equivalence relation.
Let f : X -> Y be a function. Define a relation R on X given be R = {(a, b) : f(a) = f(b)}. Show that R is an equivalence relation ?
Let A = {1, 2, 3, 4}. Let R be the equivalence relation on A × A defined by (a, b)R(c, d) if a + d = b + c. Find the equivalence class [(1,3)].
Show that the relation R in the set N × N defined by (a, b) R (c, d) if a² + d² = b² + c² is an equivalence relation.
Show that the relation R on the set Z of all integers defined by (x, y) ∈ is divisible by 3 is an equivalence relation.
Let R be the equivalence relation in the set A = {0, 1, 2, 3, 4, 5} given by R = {(a, b) : 2 divides (a –b)}. Write the equivalence class [0].
Let A = {1, 2, 3, . ...., 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d =b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)].
Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ =
Find the equation of the tangent line to the curve y =x² -2x + 7 which is
(i) parallel to the line 2x-y +9 =0
(ii) perpendicular to the line 5y - 15x = 13.
Solve the differential equation x cos x + sin x, given that y = 1 when
Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1.
Obtain the differential equation of all the circles of radius r.
Consider the experiment of tossing a coin. If the coin shows head, toss is done again, but if it shows tail, then throw a die. Find the conditional probability of the events that ‘the die shows a number greater than 4’, given that ‘there is atleast one tail’.
Show that the function f:R → R defined by f(x) = is neither one-one nor onto. Also,
if g:R → R is defined as g(x) = 2x – 1, find fog(x).